Modelling fuel economy and more economical options

Consider a situation in which two motorists, Arwa and Bao, share the same driving route but own different sized vehicles. Arwa fills up her vehicle’s tank at a station along her normal route for US$ p(1) per litre. On the other hand, Bao drives an extra d kilometres out of his normal route to fill up his vehicle’s tank for US$ p(2) per litre where p(2) < p(1) .

“Effective litres” is a method of comparing the cost of petrol bought under the two options
described above. Effective litres, for a given vehicle, are those litres used when the vehicle travels its normal route (not including the extra distance travelled in option 2).

1.) Use your variables and parameters to write algebraic expressions for E(1) and E(2) which
represent the cost per effective litre under options 1 and 2 respectively.

2.) Write a model that helps motorists decide on the more economical option for their
vehicles.

3.) Use your model to find the farthest distance that Bao should drive to obtain a 2 % price
saving.

4.) Investigate the relationship between d and p(2) when E(2) is kept constant (e.g. US$0.90,US$1.00, … etc.). Use technology to draw a family of curves for Arwa’s vehicle. Repeat for Bao’s vehicle.

5.) For E(2) = US$0.90, provide Arwa with information on three different stations that yield
this same cost per effective litre. Discuss how such information may be useful to Arwa.

6.) For E(2) = US$1.00 and p(2) = US$0.80, compare the maximum distance that each motorist should drive and still save money.

7.) Modify your model to account for the time taken to drive to and from an off-route station. Clearly justify any assumptions you make.